What is the difference between search and planning?
What is planning?
Planning (aka automated planning or AI planning) is the process of searching for a plan, which is a sequence of actions that bring the agent/world from an initial state to one or more goal states, or a policy, which a function from states to actions (or probability distributions over actions). Planning is not just the use of a search algorithm (e.g. a state-space search algorithm, such as A*) to find the plan, but planning also involves the modelling of a problem by describing it in some way, e.g. with an action language, such as PDDL or ADL, or even with propositional logic.
There are different planning problems, such as

classical planning
planning in Markov Decision Processes (which are mathematical models that describe stochastic environments/problems, i.e. environments where an action $a$ may not always lead to the same outcome)
temporal planning (i.e. when multiple actions can be taken at the same time and/or their duration may be different/variable)
hierarchical vs non-hierarchical planning

There are different planning approaches, such as

planning by state-space search (with or without heuristics), i.e. formulate the planning problem so that you can apply a state-space search algorithm  (examples of state-space search algorithms are A*, IDA*, WA*, DFS or BFS); this is probably the approach to planning that makes people confuse planning with search (which typically implicitly refers to state-space search)

planning by SAT, i.e. formulate the planning problem so that you can you a SAT solver to solve it (e.g. SATPLAN)

planning by solving constraint satisfaction problems (similar to SAT)

planning by &quot;symbolic&quot; search methods (such as Binary Decision Diagrams)


There are many different planning algorithms or planners (of course, each of these corresponds to some approach or is used to solve a particular planning problem), such as

policy iteration in the context of Markov decision processes
GraphPlan
STRIPS (which stands for Stanford Research Institute Problem Solver, but note that STRIPS also and maybe more commonly refers to the corresponding description language used to describe the planning problem)

What is search?
In the context of AI, when people use the word search, they can either implicitly refer to state-space search (with algorithms like A*) or, more vaguely/generally, to any type of search, e.g. gradient descent can also be viewed as a search algorithm, typically, in a continuous state space (rather than discrete state space), e.g. the space of real-valued vectors of parameters for a certain model (e.g. a neural network).
What is the difference between planning and search?
The difference should more or less be clear from my first paragraph above that describes planning: search is used for planning, but the planning process/study also includes other things, such as describing the problem with an action language, such as PDDL. You can also view planning as an application of search algorithms of different types. In any case, planning is about searching for plans or policies. You should probably check this course for more info. This definition of (automated) planning is consistent with our usual notion of planning.